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Chapter 4: Problem 14

Find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\). $$ t=\frac{3 \pi}{4} $$

Short Answer

Expert verified
The point on the unit circle that corresponds to \(t = \frac{3\pi}{4}\) is \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\).

Step by step solution

01

Understand the problem

The problem asks to find the point on the unit circle that corresponds to the real number \(t = \frac{3\pi}{4}\). It should be remembered that the unit circle is a circle of radius 1 centered at the origin of a coordinate plane. Each point \((x, y)\) on the unit circle is determined by falling at an angle \(t\) counter-clockwise from the positive x-axis. The x-coordinate of the point is \(\cos(t)\) and the y-coordinate is \(\sin(t)\).
02

Find the x and y coordinates

Given that \(t = \frac{3\pi}{4}\), the x-coordinate of the point is \(\cos\left(\frac{3\pi}{4}\right)\) and the y-coordinate is \(\sin\left(\frac{3\pi}{4}\right)\). By evaluating these expressions, we find that the x-coordinate is \(-\frac{1}{\sqrt{2}}\) and the y-coordinate is \(\frac{1}{\sqrt{2}}\).
03

Provide the final solution

Therefore, the point on the unit circle for \(t = \frac{3\pi}{4}\) is \(\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\).

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